The semilinear heat equation on sparse random graphs

TL;DR

This paper establishes a rigorous continuum limit for differential equations on sparse random graphs using $L^p$-graphon theory, enabling approximation of discrete models by nonlocal diffusion equations across various network types.

Contribution

It introduces a novel framework for deriving continuum limits of dynamical systems on sparse graphs, including power law networks, which are common in real-world applications.

Findings

Solutions of discrete models can be approximated by nonlocal diffusion equations.

The framework applies to diverse network topologies, including power law graphs.

Results are relevant for physical, biological, social, and economic networks.

Abstract

Using the theory of $L^p$-graphons (Borgs et al, 2014), we derive and rigorously justify the continuum limit for systems of differential equations on sparse random graphs. Specifically, we show that the solutions of the initial value problems for the discrete models can be approximated by those of an appropriate nonlocal diffusion equation. Our results apply to a range of spatially extended dynamical models of different physical, biological, social, and economic networks. Importantly, our assumptions cover network topologies featured in many important real-world networks. In particular, we derive the continuum limit for coupled dynamical systems on power law graphs. The latter is the main motivation for this work.